For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex embeddable in $\mathbb{R}^d$ has an embedding which is linear on every face?

It is true when $d \ge 2k+1$ by putting things in general position.

I am especially interested to know if anything is known about the case $(k,d)=(2,3)$.

I vaguely remember an old conjecture of Branko Grünbaum that every triangulation of the torus admitting a "straight" embedding in $\mathbb{R}^3$ but I don't know a reference (or whether this is still open).

2 Answers
2

The answer is negative for all pairs $(k,d)$ with $k+1\leq d\leq 2k$, as long as $k \ge 2$.

Brehm [1] constructed a triangulation of the Möbius strip that does not admit a geometric (simplexwise linear) embedding into $\mathbb{R}^3$.

More generally, for every pair $(k,d)$ with $k+1\leq d\leq 2k$, Brehm and Sarkaria [2] constructed an example of a $k$-dimensional simplicial complex that admits a piecewise
linear embedding into $\mathbb{R}^d$, but no geometric embedding.

Moreover, for any given integr $r \geq 0$, there is such a $K$ such that even the $r$-fold barycentric subdivision of $K$ is not geometrically embeddable into $R^d$.

Furthermore,
for certain values of the parameters, e.g., for $k=d-1$ and $d\geq 5$, it is known that
there is no recursive bound on the complexity of the subdivision needed to embed a finite
$k$-dimensional simplicial complex piecewise linearly into $\mathbb{R}^d$ (see [3, Corollary 1.2]).

"When does this embedding have a geometric realization? The problem, restricted to triangulations, was ﬁrst proposed by Grünbaum ([13], Exercise 13.2.3), who conjectured that "Every closed orientable triangulated 2-manifold without boundary can be embedded geometrically in three-dimensional Euclidean space $\mathbb{R}^3$" (see also [6]). This conjecture was recently
disproven by Bokowski and Guedes de Oliveira [4], who showed that a certain
triangulation of the complete graph $K_{12}$ on a surface of genus $6$ cannot be
realized geometrically. Brehm and Schild [5] showed that every triagulation
of the torus does have a realization in $\mathbb{R}^4$.